Finite State Machines

1
Finite State Machines

• Thursday March 20

2
Motivation where?

• Regular expressions generalized search (and
  replace)
• Editors, sed/awk, perl
• Simple parsers for languages
• Control systems of all sorts
• GUI interaction model
• Elevator, vending machine, car-building robot
• Compilers, interpreters

3
Motivation why

• Provides a simple model of computation
• Lots of nice results
• Most are easy to prove!
• But only so powerful
• Occurs a lot
• And not just in computer science!

4
What

• Informal examples (blackboard)
• Notice
• Start state
• End state(s)
• Nodes
• (labeled) transitions

5
Formal definition

• A Finite State Machine (FSM) is a tuple (Q, S,
  d, s, F) where
• Q alphabet of state symbols
• S alphabet of input symbols
• - transition
  function (total or partial)
• starting state
• final state(s)

6
Full Example

• See blackboard again

7
FSM as computation

• (see board drawing)
• FSM with input x
• String x placed on tape one letter per cell
• Read head on leftmost cell
• Current state s
• FSM started

8
FSM as computation (cont)

• Execute cycle
• Symbol under head read (current symbol). If no
  symbol, terminate.
• Compute next state (using current state and
  current symbol). If next state is undefined
  abort only occurs if state function partial
• Move head right
• Current state next state
• Repeat

9
FSM as computation (cont)

• Note that tape is read-only, one-way and finite
• Note that controller memory also finite, but
  read/write
• Additionally define the output of a FSM to be
  its last state (including undefined if necessary)

10
Languages

• Definition Let M be a FSM. We say that M
  accepts a language L (of finite strings from Ms
  input alphabet S) if and only if
• For all t in L, FSM on t ends in a final state
• For all t not in L, FSM on t does not end in a
  final state

11
Motivation revisited

• Regular expressions correspond exactly to
  languages accepted by FSM
• Most control systems are FSM!
• Exposing FSM of a GUI controller leads to much
  simpler explanation, design, coding,